过给定空间三点的最小球面

青青子衿

以过空间三点圆的任意一条直径为球面直径的球面方程
\begin{align*}
\begin{vmatrix}
\frac{\begin{vmatrix}  
{\color{red}{x^2+y^2+z^2}}&{\color{blue}y}&{\color{blue}z}&1\\  
{x_1}^2+{y_1}^2+{z_1}^2&y_1&z_1&1\\  
{x_2}^2+{y_2}^2+{z_2}^2&y_2&z_2&1\\  
{x_3}^2+{y_3}^2+{z_3}^2&y_3&z_3&1
\end{vmatrix}}{2\begin{vmatrix}  
{\color{blue}x}&{\color{blue}y}&{\color{blue}z}&1\\  
x_1&y_1&z_1&1\\  
x_2&y_2&z_2&1\\  
x_3&y_3&z_3&1
\end{vmatrix}}&

\frac{\begin{vmatrix}  
{\color{blue}x}&{\color{red}{x^2+y^2+z^2}}&{\color{blue}z}&1\\  
x_1&{x_1}^2+{y_1}^2+{z_1}^2&z_1&1\\  
x_2&{x_2}^2+{y_2}^2+{z_2}^2&z_2&1\\  
x_3&{x_3}^2+{y_3}^2+{z_3}^2&z_3&1
\end{vmatrix}}{2\begin{vmatrix}  
{\color{blue}x}&{\color{blue}y}&{\color{blue}z}&1\\  
x_1&y_1&z_1&1\\  
x_2&y_2&z_2&1\\
x_3&y_3&z_3&1
\end{vmatrix}}&\frac{\begin{vmatrix}  
{\color{blue}x}&{\color{blue}y}&{\color{red}{x^2+y^2+z^2}}&1\\  
x_1&y_1&{x_1}^2+{y_1}^2+{z_1}^2&1\\  
x_2&y_2&{x_2}^2+{y_2}^2+{z_2}^2&1\\  
x_3&y_3&{x_3}^2+{y_3}^2+{z_3}^2&1
\end{vmatrix}}{2\begin{vmatrix}  
{\color{blue}x}&{\color{blue}y}&{\color{blue}z}&1\\  
x_1&y_1&z_1&1\\  
x_2&y_2&z_2&1\\  
x_3&y_3&z_3&1
\end{vmatrix}}&1\\  
x_1&y_1&z_1&1\\  
x_2&y_2&z_2&1\\
x_3&y_3&z_3&1
\end{vmatrix} =0\end{align*}
...
\begin{align*}
\begin{vmatrix}
\begin{vmatrix}  
{\frac{\color{red}{x^2+y^2+z^2}}{2}}&{\color{blue}y}&{\color{blue}z}&1\\  
\frac{{x_1}^2+{y_1}^2+{z_1}^2}{2}&y_1&z_1&1\\  
\frac{{x_2}^2+{y_2}^2+{z_2}^2}{2}&y_2&z_2&1\\  
\frac{{x_3}^2+{y_3}^2+{z_3}^2}{2}&y_3&z_3&1
\end{vmatrix}&
\begin{vmatrix}  
{\color{blue}x}&{\frac{\color{red}{x^2+y^2+z^2}}{2}}&{\color{blue}z}&1\\  
x_1&\frac{{x_1}^2+{y_1}^2+{z_1}^2}{2}&z_1&1\\  
x_2&\frac{{x_2}^2+{y_2}^2+{z_2}^2}{2}&z_2&1\\  
x_3&\frac{{x_3}^2+{y_3}^2+{z_3}^2}{2}&z_3&1
\end{vmatrix}
&\begin{vmatrix}  
{\color{blue}x}&{\color{blue}y}&{\frac{\color{red}{x^2+y^2+z^2}}{2}}&1\\  
x_1&y_1&\frac{{x_1}^2+{y_1}^2+{z_1}^2}{2}&1\\  
x_2&y_2&\frac{{x_2}^2+{y_2}^2+{z_2}^2}{2}&1\\  
x_3&y_3&\frac{{x_3}^2+{y_3}^2+{z_3}^2}{2}&1
\end{vmatrix}
&
\begin{vmatrix}  
{\color{blue}x}&{\color{blue}y}&{\color{blue}z}&1\\  
x_1&y_1&z_1&1\\  
x_2&y_2&z_2&1\\  
x_3&y_3&z_3&1
\end{vmatrix}
\\  
x_1&y_1&z_1&1\\  
x_2&y_2&z_2&1\\
x_3&y_3&z_3&1
\end{vmatrix} =0\end{align*}

青青子衿

\begin{align*}
X^2+Y^2+Z^2&=\begin{vmatrix}   
\begin{vmatrix}   
{\color{red}{1}}&{\color{blue}0}&{\color{blue}0}&0\\   
x_1&y_1&z_1&1\\   
x_2&y_2&z_2&1\\   
x_3&y_3&z_3&1   
\end{vmatrix}&   
\begin{vmatrix}   
{\color{blue}0}&{\color{red}{1}}&{\color{blue}0}&0\\   
x_1&y_1&z_1&1\\   
x_2&y_2&z_2&1\\   
x_3&y_3&z_3&1   
\end{vmatrix}
&\begin{vmatrix}   
{\color{blue}0}&{\color{blue}0}&{\color{red}{1}}&0\\   
x_1&y_1&z_1&1\\   
x_2&y_2&z_2&1\\   
x_3&y_3&z_3&1   
\end{vmatrix}
& 0 \\   
x_1&y_1&z_1&1\\   
x_2&y_2&z_2&1\\   
x_3&y_3&z_3&1   
\end{vmatrix}\\
\\
&=\begin{vmatrix}  
1&y_{\overset{\,}1}&z_{\overset{\,}1}\\  
1&y_{\overset{\,}2}&z_{\overset{\,}2}\\  
1&y_{\overset{\,}3}&z_{\overset{\,}3}\\  
\end{vmatrix}^2   
+\begin{vmatrix}  
x_{\overset{\,}1}&1&z_{\overset{\,}1}\\  
x_{\overset{\,}2}&1&z_{\overset{\,}2}\\  
x_{\overset{\,}3}&1&z_{\overset{\,}3}\\  
\end{vmatrix} ^2   
+\begin{vmatrix}  
x_{\overset{\,}1}&y_{\overset{\,}1}&1\\  
x_{\overset{\,}2}&y_{\overset{\,}2}&1\\  
x_{\overset{\,}3}&y_{\overset{\,}3}&1\\  
\end{vmatrix}^2
\end{align*}
过空间三点的圆其圆心坐标为
\begin{align*}
x_{\overset{\,}0}=\displaystyle  
\frac{
\begin{vmatrix}
1&y_{\overset{\,}1}&z_{\overset{\,}1}\\
1&y_{\overset{\,}2}&z_{\overset{\,}2}\\
1&y_{\overset{\,}3}&z_{\overset{\,}3}\\
\end{vmatrix}  
\begin{vmatrix}
x_{\overset{\,}1}&y_{\overset{\,}1}&z_{\overset{\,}1}\\
x_{\overset{\,}2}&y_{\overset{\,}2}&z_{\overset{\,}2}\\
x_{\overset{\,}3}&y_{\overset{\,}3}&z_{\overset{\,}3}\\
\end{vmatrix}  
+
\begin{vmatrix}
x_{\overset{\,}1}&1&z_{\overset{\,}1}\\
x_{\overset{\,}2}&1&z_{\overset{\,}2}\\
x_{\overset{\,}3}&1&z_{\overset{\,}3}\\
\end{vmatrix}  
\begin{vmatrix}
\frac{{x_{\overset{\,}1}}^2+{y_{\overset{\,}1}}^2+{z_{\overset{\,}1}}^2}{2}&1&z_{\overset{\,}1}\\
\frac{{x_{\overset{\,}2}}^2+{y_{\overset{\,}2}}^2+{z_{\overset{\,}2}}^2}{2}&1&z_{\overset{\,}2}\\
\frac{{x_{\overset{\,}3}}^2+{y_{\overset{\,}3}}^2+{z_{\overset{\,}3}}^2}{2}&1&z_{\overset{\,}3}\\
\end{vmatrix}
+\begin{vmatrix}
x_{\overset{\,}1}&y_{\overset{\,}1}&1\\
x_{\overset{\,}2}&y_{\overset{\,}2}&1\\
x_{\overset{\,}3}&y_{\overset{\,}3}&1\\
\end{vmatrix}  
\begin{vmatrix}
\frac{{x_{\overset{\,}1}}^2+{y_{\overset{\,}1}}^2+{z_{\overset{\,}1}}^2}{2}&y_{\overset{\,}1}&1\\
\frac{{x_{\overset{\,}2}}^2+{y_{\overset{\,}2}}^2+{z_{\overset{\,}2}}^2}{2}&y_{\overset{\,}2}&1\\
\frac{{x_{\overset{\,}3}}^2+{y_{\overset{\,}3}}^2+{z_{\overset{\,}3}}^2}{2}&y_{\overset{\,}3}&1\\
\end{vmatrix}}  
{\begin{vmatrix}
1&y_{\overset{\,}1}&z_{\overset{\,}1}\\
1&y_{\overset{\,}2}&z_{\overset{\,}2}\\
1&y_{\overset{\,}3}&z_{\overset{\,}3}\\
\end{vmatrix}^2  
+\begin{vmatrix}
x_{\overset{\,}1}&1&z_{\overset{\,}1}\\
x_{\overset{\,}2}&1&z_{\overset{\,}2}\\
x_{\overset{\,}3}&1&z_{\overset{\,}3}\\
\end{vmatrix} ^2  
+\begin{vmatrix}
x_{\overset{\,}1}&y_{\overset{\,}1}&1\\
x_{\overset{\,}2}&y_{\overset{\,}2}&1\\
x_{\overset{\,}3}&y_{\overset{\,}3}&1\\
\end{vmatrix}^2}\\
\\
y_{\overset{\,}0}=\displaystyle  
\frac{
\begin{vmatrix}
1&y_{\overset{\,}1}&z_{\overset{\,}1}\\
1&y_{\overset{\,}2}&z_{\overset{\,}2}\\
1&y_{\overset{\,}3}&z_{\overset{\,}3}\\
\end{vmatrix}  
\begin{vmatrix}
1&\frac{{x_{\overset{\,}1}}^2+{y_{\overset{\,}1}}^2+{z_{\overset{\,}1}}^2}{2}&z_{\overset{\,}1}\\
1&\frac{{x_{\overset{\,}2}}^2+{y_{\overset{\,}2}}^2+{z_{\overset{\,}2}}^2}{2}&z_{\overset{\,}2}\\
1&\frac{{x_{\overset{\,}3}}^2+{y_{\overset{\,}3}}^2+{z_{\overset{\,}3}}^2}{2}&z_{\overset{\,}3}\\
\end{vmatrix}  
+
\begin{vmatrix}
x_{\overset{\,}1}&1&z_{\overset{\,}1}\\
x_{\overset{\,}2}&1&z_{\overset{\,}2}\\
x_{\overset{\,}3}&1&z_{\overset{\,}3}\\
\end{vmatrix}  
\begin{vmatrix}
x_{\overset{\,}1}&y_{\overset{\,}1}&z_{\overset{\,}1}\\
x_{\overset{\,}2}&y_{\overset{\,}2}&z_{\overset{\,}2}\\
x_{\overset{\,}3}&y_{\overset{\,}3}&z_{\overset{\,}3}\\
\end{vmatrix}
+\begin{vmatrix}
x_{\overset{\,}1}&y_{\overset{\,}1}&1\\
x_{\overset{\,}2}&y_{\overset{\,}2}&1\\
x_{\overset{\,}3}&y_{\overset{\,}3}&1\\
\end{vmatrix}  
\begin{vmatrix}
x_{\overset{\,}1}&\frac{{x_{\overset{\,}1}}^2+{y_{\overset{\,}1}}^2+{z_{\overset{\,}1}}^2}{2}&1\\
x_{\overset{\,}2}&\frac{{x_{\overset{\,}2}}^2+{y_{\overset{\,}2}}^2+{z_{\overset{\,}2}}^2}{2}&1\\
x_{\overset{\,}3}&\frac{{x_{\overset{\,}3}}^2+{y_{\overset{\,}3}}^2+{z_{\overset{\,}3}}^2}{2}&1\\
\end{vmatrix}}  
{\begin{vmatrix}
1&y_{\overset{\,}1}&z_{\overset{\,}1}\\
1&y_{\overset{\,}2}&z_{\overset{\,}2}\\
1&y_{\overset{\,}3}&z_{\overset{\,}3}\\
\end{vmatrix}^2  
+\begin{vmatrix}
x_{\overset{\,}1}&1&z_{\overset{\,}1}\\
x_{\overset{\,}2}&1&z_{\overset{\,}2}\\
x_{\overset{\,}3}&1&z_{\overset{\,}3}\\
\end{vmatrix} ^2  
+\begin{vmatrix}
x_{\overset{\,}1}&y_{\overset{\,}1}&1\\
x_{\overset{\,}2}&y_{\overset{\,}2}&1\\
x_{\overset{\,}3}&y_{\overset{\,}3}&1\\
\end{vmatrix}^2}\\
\\
z_{\overset{\,}0}=\displaystyle  
\frac{
\begin{vmatrix}
1&y_{\overset{\,}1}&z_{\overset{\,}1}\\
1&y_{\overset{\,}2}&z_{\overset{\,}2}\\
1&y_{\overset{\,}3}&z_{\overset{\,}3}\\
\end{vmatrix}  
\begin{vmatrix}
1&y_{\overset{\,}1}&\frac{{x_{\overset{\,}1}}^2+{y_{\overset{\,}1}}^2+{z_{\overset{\,}1}}^2}{2}\\
1&y_{\overset{\,}2}&\frac{{x_{\overset{\,}2}}^2+{y_{\overset{\,}2}}^2+{z_{\overset{\,}2}}^2}{2}\\
1&y_{\overset{\,}3}&\frac{{x_{\overset{\,}3}}^2+{y_{\overset{\,}3}}^2+{z_{\overset{\,}3}}^2}{2}\\
\end{vmatrix}  
+
\begin{vmatrix}
x_{\overset{\,}1}&1&z_{\overset{\,}1}\\
x_{\overset{\,}2}&1&z_{\overset{\,}2}\\
x_{\overset{\,}3}&1&z_{\overset{\,}3}\\
\end{vmatrix}  
\begin{vmatrix}
x_{\overset{\,}1}&1&\frac{{x_{\overset{\,}1}}^2+{y_{\overset{\,}1}}^2+{z_{\overset{\,}1}}^2}{2}\\
x_{\overset{\,}2}&1&\frac{{x_{\overset{\,}2}}^2+{y_{\overset{\,}2}}^2+{z_{\overset{\,}2}}^2}{2}\\
x_{\overset{\,}3}&1&\frac{{x_{\overset{\,}3}}^2+{y_{\overset{\,}3}}^2+{z_{\overset{\,}3}}^2}{2}\\
\end{vmatrix}
+\begin{vmatrix}
x_{\overset{\,}1}&y_{\overset{\,}1}&1\\
x_{\overset{\,}2}&y_{\overset{\,}2}&1\\
x_{\overset{\,}3}&y_{\overset{\,}3}&1\\
\end{vmatrix}  
\begin{vmatrix}
x_{\overset{\,}1}&y_{\overset{\,}1}&z_{\overset{\,}1}\\
x_{\overset{\,}2}&y_{\overset{\,}2}&z_{\overset{\,}2}\\
x_{\overset{\,}3}&y_{\overset{\,}3}&z_{\overset{\,}3}\\
\end{vmatrix}}  
{\begin{vmatrix}
1&y_{\overset{\,}1}&z_{\overset{\,}1}\\
1&y_{\overset{\,}2}&z_{\overset{\,}2}\\
1&y_{\overset{\,}3}&z_{\overset{\,}3}\\
\end{vmatrix}^2  
+\begin{vmatrix}
x_{\overset{\,}1}&1&z_{\overset{\,}1}\\
x_{\overset{\,}2}&1&z_{\overset{\,}2}\\
x_{\overset{\,}3}&1&z_{\overset{\,}3}\\
\end{vmatrix} ^2  
+\begin{vmatrix}
x_{\overset{\,}1}&y_{\overset{\,}1}&1\\
x_{\overset{\,}2}&y_{\overset{\,}2}&1\\
x_{\overset{\,}3}&y_{\overset{\,}3}&1\\
\end{vmatrix}^2}\\
\end{align*}

青青子衿

\begin{align*}
M&=  
\begin{vmatrix}  
\begin{vmatrix}   
\frac{{x_1}^2+{y_1}^2+{z_1}^2}{2}&y_1&z_1\\   
\frac{{x_2}^2+{y_2}^2+{z_2}^2}{2}&y_2&z_2\\   
\frac{{x_3}^2+{y_3}^2+{z_3}^2}{2}&y_3&z_3  
\end{vmatrix}&  
\begin{vmatrix}
x_1&\frac{{x_1}^2+{y_1}^2+{z_1}^2}{2}&z_1\\   
x_2&\frac{{x_2}^2+{y_2}^2+{z_2}^2}{2}&z_2\\   
x_3&\frac{{x_3}^2+{y_3}^2+{z_3}^2}{2}&z_3  
\end{vmatrix}
&\begin{vmatrix}     
x_1&y_1&\frac{{x_1}^2+{y_1}^2+{z_1}^2}{2}\\   
x_2&y_2&\frac{{x_2}^2+{y_2}^2+{z_2}^2}{2}\\   
x_3&y_3&\frac{{x_3}^2+{y_3}^2+{z_3}^2}{2}  
\end{vmatrix}
&
\begin{vmatrix}   
x_1&y_1&z_1\\   
x_2&y_2&z_2\\   
x_3&y_3&z_3
\end{vmatrix}
\\   
x_1&y_1&z_1&1\\   
x_2&y_2&z_2&1\\  
x_3&y_3&z_3&1  
\end{vmatrix}\\
\\
N&=\begin{vmatrix}  
1&y_{\overset{\,}1}&z_{\overset{\,}1}\\  
1&y_{\overset{\,}2}&z_{\overset{\,}2}\\  
1&y_{\overset{\,}3}&z_{\overset{\,}3}\\  
\end{vmatrix}^2   
+\begin{vmatrix}  
x_{\overset{\,}1}&1&z_{\overset{\,}1}\\  
x_{\overset{\,}2}&1&z_{\overset{\,}2}\\  
x_{\overset{\,}3}&1&z_{\overset{\,}3}\\  
\end{vmatrix} ^2   
+\begin{vmatrix}  
x_{\overset{\,}1}&y_{\overset{\,}1}&1\\  
x_{\overset{\,}2}&y_{\overset{\,}2}&1\\  
x_{\overset{\,}3}&y_{\overset{\,}3}&1\\  
\end{vmatrix}^2
\end{align*}
\begin{align*}
R^2={x_{\overset{\,}0}}^2+{y_{\overset{\,}0}}^2+{z_{\overset{\,}0}}^2+\dfrac{2M}{N}
\end{align*}